Uniqueness of Acceleration: Understanding Landau's Mechanics

[ngawang]

I'm guessing that it has to do with position and velocity being properties that are maintained by the particle. In otherwords, position and velocity are properties that accumulate but acceleration and higher derivatives are not maintained by particles but rather act on the particles. For instance if you have a particle that can only move in one direction, it accumulates distance at each instant in time and it accumulates velocity as well. Yet acceleration is not a property maintained by the particle in time. Acceleration contributes to that particles state but is not a component of that state.

This may not be of any help but it was a thought I had after reading some comments.

Very nice. Most useful idea yet. I guess thinking is not dead.

 

[Fredrik]

"This comment indicates a misunderstanding on your part. Questions about why a certain theory is a good theory can only be answered by other theories."

What theory proves this theory? And there is a deeper question brought up by this comments. A the base every system rests on a ground that is not justified by anything in that system. Science, as a theory, can not be justified by anything included in science itself. At some point there is either (1) self-justification or (2) no justification at all.

Science isn't a theory at all. Science consists of definitions of the term "theory", methods by which we can test the accuracy of a theory's predictions, and the knowledge obtained by coming up with theories and testing their accuracies. To a biologist for example, a theory is something that explains a fact. To a physicist, it's a falsifiable statement about the real world. Newton's theory of gravity is a theory in that sense. And so is Einstein's. The latter is a much better theory than the former, in the sense that it makes good predictions about a much wider range of experiments, and makes more accurate predictions about everything that Newton's theory makes good predictions about. It also explains why Newton's law of gravity can be used successfully in certain situations.

This sort of explanation, an answer provided by a better theory, is the best kind of answer to your question that you can hope for. WannabeNewton's approach is also good. It shows that versions of classical mechanics with higher order derivatives has problems that make them easy to rule out by experiments that have already been performed.

 

[Fredrik]

I'm guessing that it has to do with position and velocity being properties that are maintained by the particle. In otherwords, position and velocity are properties that accumulate but acceleration and higher derivatives are not maintained by particles but rather act on the particles. For instance if you have a particle that can only move in one direction, it accumulates distance at each instant in time and it accumulates velocity as well. Yet acceleration is not a property maintained by the particle in time. Acceleration contributes to that particles state but is not a component of that state.

The only reason why it seems this way is that the acceleration is determined by a differential equation of the form $x''(t)=mF(x(t),x'(t),t)$ which has a unique solution for each initial condition on the position and velocity. This is why we can think of position and velocity as "input" and acceleration as "output". So you are using the fact that you want to explain to explain the fact you want to explain.

This is not the way to explain why we don't find forces that depend on higher derivatives in nature.

 

[Fredrik]

See this video on "why" questions:

I just watched this video. It's pretty great. It's impressive that he could come up with such a good answer in an interview. He had probably used some of those ideas before, but still. I like how he at the end of the video leaned back and smiled, as if he was thinking "yeah that was an awesome answer".

 

[WannabeNewton]

This is one of the better trolls we have had in a long time I must say.

 

[micromass]

This is one of the better trolls we have had in a long time I must say.

Dunno, that guy from yesterday with his numerology was pretty good.

 

[WannabeNewton]

Dunno, that guy from yesterday with his numerology was pretty good.

Yeah but he was too polite. What kind of troll is polite?

 

[berkeman]

This [STRIKE]is[/STRIKE] was one of the better trolls we have had in a long time I must say.

I corrected your post.

 

[Jano L.]

It is an experimental result.

You can invent a framework where the third derivative of the position depends on position, velocities and accelerations, where you have to know all three to get full knowledge about the system. According to all measurements done so far, we do not live in such a universe-

I think this was a nice answer. In other words, we use 2nd order equations with 2 initial conditions and not more because that is the simplest framework that works well in practice. We even call it "Newton's laws", but this does not prevent the possibility that some motions are out of scope of such theory, we just did not observe such motions.

Parenthetically. Even in the framework of 2nd order differential equations, position and velocity is not always sufficient. Consider motion of particle in the potential

$$
U(x) = -\frac{|x|^{3/2}}{3/2}
$$
with initial conditions
$$
x(0) = 0
$$
$$
\dot{x}(0) = 0
$$
and equation of motion
$$
m\ddot{x} = -\partial U/\partial x.
$$

This has infinity of solutions

$$
x(t) = 0,
$$
or
$$
x(t)=\left(\frac{1}{12m}\right)^2(t-t_0)^4 \theta(t-t_0),
$$
for any $t_0>0$. Which one is realized can be determined if we know the value of $d^4x/dt^4(t)$ in addition.

 

[lightarrow]

In Landau's Mechanics it states "If all co-ordinates and velocities are simultaneously specified, it is know from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically, this means that, if all the co-ordinates q and velocities \dot{q} are given at some instant, the accelerations \ddot{q} at that instant are uniquely defined."

My question is why is this so. I understand that from knowing the co-ordinates of a mechancial system the future evolution of a system is not uniquely determined. But how does the additional knowledge of the velocities uniquely determine the acceleration of the system and hence its future mechanical state?

It is sincerely very difficult to understand what you intended to ask in this thread, so I can try with the following.

We know from 2° Newton law that:

m\ddot{x} = F where F is the force field. You seem to be asking why F depends on {x}, \dot{x}, t only, and not even on \ddot{x}for example. So let's assume a force field F' depended on the second derivative of the position, so that Newton's law would be:m\ddot{x} = F'({x},\dot{x},\ddot{x},t)But this would still be a second order differential equation on the unknown x(t), so its solution would depend again from 2 initial conditions only, for example, the usual:x(0),\dot{x}(0).